In the previous post, we showed that is a flat map, and stated the Hopkins-Miller theorem, namely that there exists a sheaf of -rings on . The global sections of this sheaf is an -ring spectrum .
As we’ll be using the term ‘‘derived stack’’ a lot, it’s probably a smart idea to present a definition.
Definition: A derived stack is a Deligne-Mumford stack along with a sheaf of -rings on the etale site of such that is the structure sheaf of (which, recall, is defined by the colimit of the structure sheaves of the affine etales over ). We’ll call a ‘‘derived lift’’ of .
Lurie’s theorem provides a slick method of derived stacks. Let’s recall his theorem in the generality stated in the Behrens-Lawson book on topological automorphic forms.
Theorem (Lurie): Let be a local ring in mixed characteristic, and let be a locally Noetherian separated Deligne-Mumford stack over . Let be a -divisible group of height and dimension , and let denote the moduli of -divisible groups of dimension and height . If the associated map is formally etale, or, equivalently, if the following condition is satisfied:
- for some etale cover , and for every , the map classifying the deformation of is an isomorphism of formal schemes
then there is an even periodic presheaf of -rings on that is a sheaf in the etale topology that makes a derived stack.
The reason this theorem was proved was to provide another, more canonical, in some sense, construction of the derived moduli stack of elliptic curves. Indeed, we get a sheaf of -rings on (note that is the same as the lift of constructed by Hopkins-Miller, because Lurie also asserts that the space of derived lifts of is path-connected and has a preferred basepoint) from the following theorem of Serre-Tate.
Theorem (Serre-Tate): The deformation theory of an elliptic curve is controlled by the deformation theory of its corresponding -divisible group . In other words, the map is formally etale.
There is a more tautological example, which the theorem’s statement is clearly trying to emulate. In particular, we can consider the Lubin-Tate space, which parametrizes deformations of formal groups, in the sense that if is a perfect field and a formal group of height over , there is a bijection
By Lubin-Tate theory, we know that is a formal affine scheme over , and there is a (noncanonical!) isomorphism
where the are defined by – this is why they are noncanonical – where denotes the universal deformation of . The ring is a complete regular local Noetherian ring, with maximal ideal . That’s a regular sequence, and so in particular, there is a homotopy commutative even periodic Landweber exact ring spectrum (I’ll suppress the from the notation) such that , where . This is the famous Morava E-theory at the height (and the prime ).
When , for instance, this coefficient ring is , and is -adic complex -theory. However – and this is confusing – the formal group laws are not the same. The formal group law associated to -adic complex -theory is the multiplicative group law , while the formal group law associated to is the -typification of the multiplicative formal group law.
Anyway, we get a derived lift of from the following theorem of Goerss-Hopkins-Miller.
Theorem (Goerss-Hopkins-Miller): The homotopy commutative ring spectrum is an -ring spectrum.
The structure sheaf is straightforward to construct: given an etale cover , define by using Lurie’s theorem that the affine etale site of is the same as the affine etale site of . This derived stack is denoted .